Regular sequences and local cohomology modules with respect to a pair of ideals
aa r X i v : . [ m a t h . A C ] M a y REGULAR SEQUENCES AND LOCAL COHOMOLOGY MODULESWITH RESPECT TO A PAIR OF IDEALS
SH. PAYROVI AND M. LOTFI PARSAA
BSTRACT . Let R be a Noetherian ring, I and J two ideals of R and t an inte-ger. Let S be the class of Artinian R -modules, or the class of all R -modules N with dim R N ≤ k , where k is an integer. It is proved that inf { i : H iI , J ( M ) / ∈ S } = inf { S − depth a ( M ) : a ∈ e W ( I , J ) } , where M is a finitely generated R -module, oris a ZD -module such that M / a M / ∈ S for all a ∈ e W ( I , J ) . Let Supp R H iI , J ( M ) bea finite subset of Max ( R ) for all i < t . It is shown that there are maximal ideals m , m , . . . , m k of R such that H iI , J ( M ) ∼ = H i m ( M ) ⊕ H i m ( M ) ⊕ · · · ⊕ H i m k ( M ) forall i < t .
1. I
NTRODUCTION
Throughout this paper, R is a commutative Noetherian ring with non-zero iden-tity, I and J are two ideals of R , M is an R -module and s and t are two integers. Fornotations and terminologies not given in this paper, the reader is referred to [4], [5]and [16] if necessary.The theory of local cohomology, which was introduced by Grothendieck [10], isa useful tool for attacking problems in commutative algebra and algebraic geom-etry. Bijan-Zadeh [3] introduced the local cohomology modules with respect to asystem of ideals, which is a generalization of ordinary local cohomology modules.As a special case of these extend modules, Takahashi, Yoshino and Yoshizawa [16]defined the local cohomology modules with respect to a pair of ideals. To be moreprecise, let W ( I , J ) = { p ∈ Spec ( R ) : I t ⊆ J + p for some positive integer t } . Theset of elements x of M such that Supp R Rx ⊆ W ( I , J ) , is said to be ( I , J ) -torsionsubmodule of M and is denoted by G I , J ( M ) . G I , J ( − ) is a covariant, R -linear functorfrom the category of R -modules to itself. For an integer i , the local cohomologyfunctor H iI , J ( − ) with respect to ( I , J ) , is defined to be the i -th right derived functorof G I , J ( − ) . Also H iI , J ( M ) is called the i -th local cohomology module of M with re-spect to ( I , J ) . If J =
0, then H iI , J ( − ) coincides with the ordinary local cohomology ZD -module. functor H iI ( − ) . Let e W ( I , J ) = { a E R : I t ⊆ J + a for some positive integer t } . It iseasy to see that G I , J ( M ) = { x ∈ M : ∃ a ∈ e W ( I , J ) , a x = } = [ a ∈ e W ( I , J ) ( M a ) . In section 2, we study extension functors of local cohomology modules withrespect to a pair of ideals. Let S be a Melkersson subcategory with respect to I ,and M a finitely generated R -module. The current authors, in [13, Theorem 2.11],showed that if H iI , J ( M ) ∈ S for all i < t , then H iI ( M ) ∈ S for all i < t . In 2.5, weimprove this result and we show that if Ext jR ( N , H iI , J ( M )) ∈ S for all i < t and all j < t − i , then H i a ( M ) ∈ S for all i < t , where M is an arbitrary R -module and N is afinitely generated R -module with Supp R N = V ( a ) for some a ∈ e W ( I , J ) .Let S be a Serre subcategory of the category of R -modules. Aghapournahr andMelkersson [1] introduced the notion of S -sequences on M as a generalization ofregular sequences. Suppose that S is a Melkersson subcategory with respect to I , M is a ZD -module and M / IM / ∈ S . In [14, Theorem 2.9] it is proved that all maximal S -sequences on M in I , have the same length. If this common length is denotedby S − depth I ( M ) , then S − depth I ( M ) = inf { i : H iI ( M ) / ∈ S } ; see [14, Corollary2.12]. In 2.25, we generalize this result as follows. Let S be the class of Artinian R -modules, or the class of all R -modules N with dim R N ≤ k , where k is an integer.Then inf { S − depth a ( M ) : a ∈ e W ( I , J ) } = inf { i : H iI , J ( M ) / ∈ S } , where M is a finitelygenerated R -module, or is a ZD -module such that M / a M / ∈ S for all a ∈ e W ( I , J ) .In section 3, we get some identities between local cohomology modules. LetSupp R H iI , J ( M ) be a finite subset of Max ( R ) for all i < t . Then there are maximalideals m , m , . . . , m k of R such that H iI , J ( M ) ∼ = H i m ( M ) ⊕ H i m ( M ) ⊕ · · · ⊕ H i m k ( M ) for all i < t ; see 3.7. As a consequence we conclude that, if ( R , m ) is a local ring,then inf { i : H iI , J ( M ) is not Artinian } = inf { i : H iI , J ( M ) = H i m ( M ) } ; see 3.10.2. E XTENSION FUNCTORS OF LOCAL COHOMOLOGY MODULES
Recall that R is a Noetherian ring, I and J are ideals of R and M is an R -module. Definition 2.1.
A full subcategory of the category of R -modules is said to be Serresubcategory, if it is closed under taking submodules, quotients and extensions. ASerre subcategory S is said to be a Melkersson subcategory with respect to I , if for EGULAR SEQUENCES AND LOCAL COHOMOLOGY MODULES... 3 any I -torsion R -module M , 0 : M I ∈ S implies that M ∈ S . A Serre subcategory iscalled Melkersson subcategory when it is a Melkersson subcategory with respect toall ideals of R .The class of finitely generated modules and the class of weakly Laskerian mod-ules are Serre subcategories. Aghapournahr and Melkersson [1, Lemma 2.2] provedthat if a Serre subcategory is closed under taking injective hulls, then it is a Melk-ersson subcategory. The class of zero modules, Artinian R -modules, modules withfinite support and the class of R -modules N with dim R N ≤ k , where k is a non-negative integer, are Serre subcategories closed under taking injective hulls, andhence are Melkersson subcategories; see [1, Example 2.4]. The class of I -cofiniteArtinian modules is a Melkersson subcategory with respect to I , but is not closedunder taking injective hulls; see [1, Example 2.5].The following result is a generalization of [2, Theorem 2.1]. Theorem 2.2.
Let N be an ( I , J ) -torsion R-module. If Ext t − iR ( N , H iI , J ( M )) ∈ S forall i ≤ t, then Ext tR ( N , M ) ∈ S.Proof.
Let F ( − ) = Hom R ( N , − ) and G ( − ) = G I , J ( − ) . Then we have FG ( M ) = Hom R ( N , M ) . By [15, Theorem 11.38], there is the Grothendieck spectral sequence E p , q : = Ext pR ( N , H qI , J ( M )) ⇒ Ext p + qR ( N , M ) . There is a finite filtration0 = j t + H t ⊆ j t H t ⊆ · · · ⊆ j H t ⊆ j H t = Ext tR ( N , M ) such that E t − i , i ¥ ∼ = j t − i H t / j t + − i H t for all i ≤ t . It is enough to show that j H t ∈ S .By hypothesis, E t − i , ij ∈ S for all j ≥ i ≤ t , and so E t − i , i ¥ ∈ S for all i ≤ t . Thesequence 0 −→ j t + − i H t −→ j t − i H t −→ E t − i , i ¥ −→ i ≤ t . Therefore it follows that j H t ∈ S . (cid:3) Corollary 2.3.
Let N be an ( I , J ) -torsion R-module. If Ext jR ( N , H iI , J ( M )) ∈ S for alli < t and all j < t − i, then Ext iR ( N , M ) ∈ S for all i < t. Corollary 2.4.
Suppose that S is a Melkersson subcategory with respect to I, andN is a finitely generated R-module with
Supp R N = V ( a ) for some a ∈ e W ( I , J ) . If SH. PAYROVI AND M. LOTFI PARSA
Ext jR ( N , H iI , J ( M )) ∈ S for all i < t and all j < t − i, then H i a ( L , M ) ∈ S for all i < tand all finitely generated R-modules L.Proof. The result follows by 2.3 and [1, Theorem 2.9]. (cid:3)
The following result improves [13, Theorem 2.11].
Corollary 2.5.
Suppose that S is a Melkersson subcategory with respect to I, andN is a finitely generated R-module with
Supp R N = V ( a ) for some a ∈ e W ( I , J ) . If Ext jR ( N , H iI , J ( M )) ∈ S for all i < t and all j < t − i, then H i a ( M ) ∈ S for all i < t.Proof. In 2.4, put L = R . (cid:3) Corollary 2.6.
If S is a Melkersson subcategory with respect to I, then inf { i : H iI , J ( M ) / ∈ S } ≤ inf { inf { i : H i a ( M ) / ∈ S } : a ∈ e W ( I , J ) } . As a generalization of finitely generated modules, Evans [9] introduced ZD -modules as follows. Definition 2.7. An R -module M is said to be zero-divisor module ( ZD -module), iffor any submodule N of M , the set Z R ( M / N ) is a finite union of prime ideals inAss R M / N .According to [7, Example 2.2], the class of ZD -modules contains finitely gen-erated, Laskerian, weakly Laskerian, linearly compact and Matlis reflexive mod-ules. Also it contains modules whose quotients have finite Goldie dimension andmodules with finite support, in particular Artinian modules. Therefore the class of ZD -modules is much larger than that of finitely generated modules. Definition 2.8.
An element a of R is called S -regular on M , if 0 : M a ∈ S . A se-quence a , . . . , a t is an S -sequence on M , if a i is S -regular on M / ( a , . . . , a i − ) M for i = , . . . , t . The S -sequence a , . . . , a t is said to be maximal S -sequence on M , if a , . . . , a t , y is not an S -sequence on M for any y ∈ R .When S is the class of zero modules, Artinian R -modules, modules with finitesupport, and the class of R -modules N with dim R N ≤ k , where k is a non-negativeinteger, then S -sequences on M are, poor M-sequences, filter-regular sequences, EGULAR SEQUENCES AND LOCAL COHOMOLOGY MODULES... 5 generalized regular sequences, and M -sequences in dimension > k , respectively;see [1, Example 2.8].Let S be a Melkersson subcategory with respect to I , and M a ZD -module suchthat M / IM / ∈ S . The current authors, in [14, Theorem 2.9], proved that all maximal S -sequences on M in I , have the same length. Definition 2.9.
Let S be a Melkersson subcategory with respect to I , and M a ZD -module such that M / IM / ∈ S . The common length of all maximal S -sequences on M in I , is denoted by S − depth I ( M ) . If M / IM ∈ S , we set S − depth I ( M ) = ¥ .Suppose that M is a ZD -module. When S is the class of zero modules, Artinian R -modules, and modules with finite support, then S − depth I ( M ) is the same asordinary depth I ( M ) , f − depth I ( M ) (filter-depth), and g − depth I ( M ) (generalizeddepth), respectively. Corollary 2.10.
Let S be a Melkersson subcategory with respect to I, and M aZD-module. Then inf { i : H iI , J ( M ) / ∈ S } ≤ inf { S − depth a ( M ) : a ∈ e W ( I , J ) } . Proof.
The result follows by 2.6 and [14, Corollary 2.12]. (cid:3)
In the following, we study the relation between generalized local cohomologymodules and local cohomology modules with respect to a pair of ideals.
Corollary 2.11.
Suppose that N is a finitely generated a -torsion R-module for some a ∈ e W ( I , J ) . If Ext t − iR ( N , H iI , J ( M )) ∈ S for all i ≤ t, then H t a ( N , M ) ∈ S.Proof.
The result follows by 2.2. Note that G a ( N ) ⊆ G I , J ( N ) and by [8, Lemma 2.1]we have Ext iR ( N , M ) ∼ = H i a ( N , M ) for any integer i . (cid:3) Corollary 2.12.
Suppose that N is a finitely generated a -torsion R-module for some a ∈ e W ( I , J ) . If Ext jR ( N , H iI , J ( M )) ∈ S for all i < t and all j < t − i, then H i a ( N , M ) ∈ Sfor all i < t. The following result is a generalization of [2, Theorem 2.3].
Theorem 2.13.
Let N be an ( I , J ) -torsion R-module. If Ext s + t + − iR ( N , H iI , J ( M )) ∈ Sfor all i < t, Ext s + t − − iR ( N , H iI , J ( M )) ∈ S for all t < i < s + t, and Ext s + tR ( N , M ) ∈ S,then
Ext sR ( N , H tI , J ( M )) ∈ S. SH. PAYROVI AND M. LOTFI PARSA
Proof.
Let F ( − ) = Hom R ( N , − ) and G ( − ) = G I , J ( − ) . Then we have FG ( M ) = Hom R ( N , M ) . By [15, Theorem 11.38], there is the Grothendieck spectral sequence E p , q : = Ext pR ( N , H qI , J ( M )) ⇒ Ext p + qR ( N , M ) . There is a finite filtration0 = j s + t + H s + t ⊆ j s + t H s + t ⊆ · · · ⊆ j H s + t ⊆ j H s + t = Ext s + tR ( N , M ) such that E s + t − i , i ¥ ∼ = j s + t − i H s + t / j s + t + − i H s + t for all i ≤ s + t . It is enough to showthat E s , t ∈ S . We have the following exact sequences0 −→ Ker d s , tt + − i −→ E s , tt + − i d s , tt + − i −→ E s + t + − i , it + − i and 0 −→ Im d s − t − + i , t − it + − i −→ Ker d s , tt + − i −→ E s , tt + − i −→ i . By hypothesis, E s + t + − i , it + − i ∈ S for all i < t , and E s + t − − i , i + i − t ∈ S for all t < i < s + t . It follows that E s − t − + i , t − it + − i ∈ S for all t − s < i < t . Note that if i ≤ t − s , then E s − t − + i , t − it + − i =
0. Hence E s − t − + i , t − it + − i ∈ S for all i < t , and there-fore Im d s − t − + i , t − it + − i ∈ S for all i < t . Also we have E s , ts + t + = E s , t ¥ ∈ S , because E s , t ¥ ∼ = j s H s + t / j s + H s + t and j s H s + t ⊆ j H s + t = Ext s + tR ( N , M ) ∈ S . Now the claimfollows by the above exact sequences. (cid:3) Corollary 2.14.
Let N be an ( I , J ) -torsion R-module. Let Ext j − iR ( N , H iI , J ( M )) ∈ Sfor j = s + t , s + t + and all i < t, and Ext s + t − − iR ( N , H iI , J ( M )) ∈ S for all t < i < s + t. Then Ext s + tR ( N , M ) ∈ S if and only if
Ext sR ( N , H tI , J ( M )) ∈ S.Proof.
The claim follows by 2.2 and 2.13. (cid:3)
The following result is a generalization of [17, Theorem 2.3].
Corollary 2.15.
Let N be an ( I , J ) -torsion R-module. If Ext t + − iR ( N , H iI , J ( M )) ∈ Sfor all i < t, and Ext tR ( N , M ) ∈ S, then
Hom R ( N , H tI , J ( M )) ∈ S.Proof.
In 2.13, put s = (cid:3) Proposition 2.16.
Let
Ext t + − iR ( R / a , H iI , J ( M )) be Artinian for all a ∈ e W ( I , J ) andall i < t, and Ext tR ( R / a , M ) be Artinian for all a ∈ e W ( I , J ) . Then Ext jR ( R / a , H tI , J ( M )) is Artinian for all a ∈ e W ( I , J ) and all j ∈ N . EGULAR SEQUENCES AND LOCAL COHOMOLOGY MODULES... 7
Proof.
By 2.15, Hom R ( R / a , H tI , J ( M )) is Artinian for all a ∈ e W ( I , J ) . Also we knowthat H tI , J ( M ) = [ a ∈ e W ( I , J ) ( H tI , J ( M ) a ) = [ a ∈ e W ( I , J ) Hom R ( R / a , H tI , J ( M )) . Now the claim follows by [11, Theorem 5.1] and [1, Theorem 2.9]. (cid:3)
Corollary 2.17.
Let
Ext iR ( R / a , M ) be Artinian for all a ∈ e W ( I , J ) and all i < t. Then Ext jR ( R / a , H iI , J ( M )) is Artinian for all a ∈ e W ( I , J ) , all i < t and all j ∈ N .Proof. It follows by 2.16 that Ext jR ( R / a , G I , J ( M )) is Artinian for all a ∈ e W ( I , J ) andall j ∈ N . Now again using of 2.16, implies that Ext jR ( R / a , H I , J ( M )) is Artinian forall a ∈ e W ( I , J ) and all j ∈ N . By continuing this process, the claim follows. (cid:3) Lemma 2.18.
If H t − iI ( H iI , J ( M )) ∈ S for all i ≤ t, then H tI , J ( M ) ∈ S.Proof.
Let F ( − ) = G I ( − ) and G ( − ) = G I , J ( − ) . Then FG ( M ) = G I , J ( − ) . The restof the proof is similar to that of 2.2. (cid:3) Corollary 2.19. If Ext iR ( R / a , M ) is Artinian for all a ∈ e W ( I , J ) and all i < t, thenH iI , J ( M ) is Artinian for all i < t.Proof. It follows by 2.17 and [1, Theorem 2.9] that H jI ( H iI , J ( M )) is Artinian for all i < t and all j ∈ N . Now the claim follows by 2.18. (cid:3) Proposition 2.20.
Let S be the class of all R-modules N with dim R N ≤ k, where kis an integer. Let Ext t + − iR ( R / a , H iI , J ( M )) ∈ S for all a ∈ e W ( I , J ) and all i < t, and Ext tR ( R / a , M ) ∈ S for all a ∈ e W ( I , J ) . Then H tI , J ( M ) ∈ S.Proof.
By 2.15, Hom R ( R / a , H tI , J ( M )) ∈ S for all a ∈ e W ( I , J ) . Also we know that H tI , J ( M ) = [ a ∈ e W ( I , J ) ( H tI , J ( M ) a ) = [ a ∈ e W ( I , J ) Hom R ( R / a , H tI , J ( M )) . It follows that H tI , J ( M ) ∈ S . (cid:3) Corollary 2.21.
Let S be the class of all R-modules N with dim R N ≤ k, where k isan integer. If Ext iR ( R / a , M ) ∈ S for all a ∈ e W ( I , J ) and all i < t, then H iI , J ( M ) ∈ Sfor all i < t. SH. PAYROVI AND M. LOTFI PARSA
Proof.
We know that G I , J ( M ) = [ a ∈ e W ( I , J ) Hom R ( R / a , G I , J ( M )) = [ a ∈ e W ( I , J ) Hom R ( R / a , M ) . Therefore G I , J ( M ) ∈ S . It follows by 2.20 that H I , J ( M ) ∈ S . By keeping this process,the claim follows. (cid:3) Corollary 2.22.
Let S be the class of Artinian R-modules, or the class of all R-modules N with dim R N ≤ k, where k is an integer. If H i a ( M ) ∈ S for all a ∈ e W ( I , J ) and all i < t, then H iI , J ( M ) ∈ S for all i < t.Proof. The claim follows by 2.19, 2.21 and [1, Theorem 2.9]. (cid:3)
Corollary 2.23.
Let S be the class of Artinian R-modules, or the class of all R-modules N with dim R N ≤ k, where k is an integer. Then the following statementsare equivalent: (i) H iI , J ( M ) ∈ S for all i < t; (ii) H i a ( M ) ∈ S for all a ∈ e W ( I , J ) and all i < t.Proof. The claim follows by 2.5 and 2.22. (cid:3)
Corollary 2.24.
Let S be the class of Artinian R-modules, or the class of all R-modules N with dim R N ≤ k, where k is an integer. Then inf { i : H iI , J ( M ) / ∈ S } = inf { inf { i : H i a ( M ) / ∈ S } : a ∈ e W ( I , J ) } . The following result is a generalization of [14, Theorem 2.13].
Theorem 2.25.
Let S be the class of Artinian R-modules, or the class of all R-modules N with dim R N ≤ k, where k is an integer. Let M be a finitely generatedR-module, or be a ZD-module such that M / a M / ∈ S for all a ∈ e W ( I , J ) . Then inf { i : H iI , J ( M ) / ∈ S } = inf { S − depth a ( M ) : a ∈ e W ( I , J ) } . Proof.
The claim follows by 2.24 and [14, Theorem 2.13]. (cid:3)
EGULAR SEQUENCES AND LOCAL COHOMOLOGY MODULES... 9
3. S
OME IDENTITIES BETWEEN LOCAL COHOMOLOGY MODULES
Suppose that E • ( M ) : 0 −→ E R ( M ) d −→ E R ( M ) −→ · · · −→ E iR ( M ) d i −→ E i + R ( M ) −→ · · · ( ∗ ) is a minimal injective resolution of M , where E iR ( M ) ∼ = L p ∈ Spec ( R ) m i ( p , M ) E R ( R / p ) is a decomposition of E iR ( M ) as the direct sum of indecomposable injective R -modules, E R ( R / p ) denotes the injective hull of R / p and m i ( p , M ) denotes the i -thBass number of M with respect to p . It follows by [16, Proposition 1.11] that G I , J ( E iR ( M )) ∼ = M p ∈ W ( I , J ) m i ( p , M ) E R ( R / p ) . Hence Supp R G I , J ( E iR ( M )) = { p ∈ W ( I , J ) : m i ( p , M ) = } . The above mentioned results are assumed known through this section.
Theorem 3.1.
Let S be a Serre subcategory closed under taking injective hulls. Thefollowing conditions are equivalent: (i) H iI , J ( M ) ∈ S for all i < t. (ii) G I , J ( E iR ( M )) ∈ S for all i < t.Proof. Since G I , J ( E iR ( M )) is injective and Ker G I , J ( d i ) = Ker d i ∩ G I , J ( E iR ( M )) , thus G I , J ( E iR ( M )) is injective hull of Ker G I , J ( d i ) . Now the claim follows by [12, Lemma5.4]. We note that the proof of [12, Lemma 5.4] is still valid if the class of Artinian R -modules is replaced by a Serre subcategory that is closed under taking injectivehulls. (cid:3) Corollary 3.2.
The following statements are equivalent: (i) Supp R H iI , J ( M ) is a finite subset of Max ( R ) for all i < t; (ii) H iI , J ( M ) is Artinian for all i < t. Corollary 3.3. If ( R , m ) is a local ring, then inf { i : Supp R H iI , J ( M )
6⊆ { m }} = inf { i : H iI , J ( M ) is not Artinian } . Proposition 3.4.
Let M be a finitely generated R-module, or be a ZD-module suchthat M p / p M p = for all p ∈ W ( I , J ) . Then inf { i : H iI , J ( M ) = } = inf { depth M p : p ∈ W ( I , J ) } . Proof.
Let t = inf { depth M p : p ∈ W ( I , J ) } . It follows by [14, Corollary 2.14] that m i ( p , M ) = p ∈ W ( I , J ) and all i < t . So G I , J ( E iR ( M )) = i < t ,and hence H iI , J ( M ) = i < t . Therefore t ≤ inf { i : H iI , J ( M ) = } . Nowit is enough to show that H tI , J ( M ) =
0. By assumption, there is q ∈ W ( I , J ) suchthat t = depth M q . It follows by [14, Corollary 2.14] that m t ( q , M ) =
0. Therefore G I , J ( E tR ( M )) =
0, and hence H tI , J ( M ) = (cid:3) We can get a generalization of [6, Theorem 2.4].
Proposition 3.5.
Let M be a finitely generated R-module, or be a ZD-module suchthat M p / p M p = for all p ∈ W ( I , J ) − Max ( R ) . Then inf { i : Supp R H iI , J ( M ) Max ( R ) } = inf { depth M p : p ∈ W ( I , J ) − Max ( R ) } . Proof.
Let t = inf { i : Supp R H iI , J ( M ) Max ( R ) } . It follows by 3.1 that m i ( p , M ) = p ∈ W ( I , J ) − Max ( R ) and all i < t , and there is q ∈ W ( I , J ) − Max ( R ) suchthat m t ( q , M ) =
0. Now it follows by [14, Corollary 2.14] that depth M p ≥ t for all p ∈ W ( I , J ) − Max ( R ) , and depth M q = t . Therefore inf { depth M p : p ∈ W ( I , J ) − Max ( R ) } = t . (cid:3) Theorem 3.6.
Let
Supp R H iI , J ( M ) be a finite subset of Max ( R ) for all i < t. Thenthere are maximal ideals m , m , . . . , m k ∈ W ( I , J ) such that H iI , J ( M ) ∼ = H i m m ··· m k ( M ) for all i < t.Proof. It follows by 3.1 that Supp R G I , J ( E i ( M )) is a finite subset of Max ( R ) for all i < t . Let Supp R G I , J ( E i ( M )) = { m i , m i , ..., m ik i } , where k i is an integer. Then G I , J ( E i ( M )) ∼ = k i M j = m i ( m i j , M ) E R ( R / m i j ) for all i < t . Put a = (cid:213) i , j m i j . Then V ( a ) = { m i j : 0 ≤ i < t , ≤ j ≤ k i } , and a ∈ e W ( I , J ) . Therefore G a ( E i ( M )) ∼ = M p ∈ V ( a ) m i ( p , M ) E R ( R / p )= k i M j = m i ( m i j , M ) E R ( R / m i j ) ∼ = G I , J ( E i ( M )) for all i < t , and hence H iI , J ( M ) ∼ = H i a ( M ) for all i < t . (cid:3) EGULAR SEQUENCES AND LOCAL COHOMOLOGY MODULES... 11
Corollary 3.7.
Let
Supp R H iI , J ( M ) be a finite subset of Max ( R ) for all i < t. Thenthere are maximal ideals m , m , . . . , m k ∈ W ( I , J ) such thatH iI , J ( M ) ∼ = H i m ( M ) ⊕ H i m ( M ) ⊕ · · · ⊕ H i m k ( M ) for all i < t.Proof. The claim follows by 3.6 and the Mayer-Vietoris sequence [4, 3.2.3]. (cid:3)
Corollary 3.8.
Let ( R , m ) be a local ring. If Supp R H iI , J ( M ) ⊆ { m } for all i < t,then H iI , J ( M ) ∼ = H i m ( M ) for all i < t. Corollary 3.9. If ( R , m ) is a local ring, then inf { i : Supp R H iI , J ( M )
6⊆ { m }} = inf { i : H iI , J ( M ) = H i m ( M ) } . The following result is a generalization of [6, Proposition 2.5].
Corollary 3.10. If ( R , m ) is a local ring, then inf { i : H iI , J ( M ) is not Artinian } = inf { i : H iI , J ( M ) = H i m ( M ) } . Proof.
The claim follows by 3.3 and 3.9. (cid:3)
Corollary 3.11.
Let ( R , m ) be a local ring. If Supp R H iI , J ( M ) ⊆ { m } for all i < t,then H iI , J ( M ) ∼ = H i a ( M ) for all a ∈ e W ( I , J ) and all i < t.Proof. It follows by 2.5 that Supp R H i a ( M ) ⊆ { m } for all a ∈ e W ( I , J ) and all i < t .Therefore H i a ( M ) ∼ = H i m ( M ) for all a ∈ e W ( I , J ) and all i < t , by 3.8. Now againusing of 3.8 implies that H iI , J ( M ) ∼ = H i m ( M ) for all i < t , and the claim follows. (cid:3) R EFERENCES [1] M. Aghapournahr, L. Melkersson,
Local cohomology and Serre subcategories , J. Algebra (2008), 1275-1287.[2] M. Aghapournahr, A. J. Taherizadeh, A. Vahidi,
Extension functors of local cohomology mod-ules , Bull. Iranian Math. Soc. (2011), 117-134.[3] M. H. Bijan-Zadeh, Torsion theories and local cohomology over commutative Noetherianrings , J. London Math. Soc. (2) (1979), 402-410.[4] M. P. Brodmann, R. Y. Sharp, Local Cohomology: An Algebraic Introduction with GeometricApplications , Cambridge University Press, Cambridge, 1998.[5] W. Bruns, J. Herzog,
Cohen-Macaulay Rings , Cambridge University Press, Cambridge, 1993. [6] L. Chu, Q. Wang,
Some results on local cohomology modules defined by a pair of ideals , J.Math. Kyoto Univ. (2009), 193-200.[7] K. Divaani-Aazar, M. A. Esmkhani, Artinianness of local cohomology modules of ZD-modules , Comm. Algebra (2005), 2857-2863.[8] K. Divaani-Aazar, R. Sazeedeh, M. Tousi, On vanishing of generalized local cohomologymodules , Algebra Colloq. (2005), 213-218.[9] E. G. Evans, Zero divisors in Noetherian-like rings , Trans. Amer. Math. Soc. (1971),505-512.[10] A. Grothendieck (notes by R. Hartshorne),
Local Cohomology, Springer Lecture Notes inMath., 41 , Springer-Verlag, 1966.[11] L. Melkersson,
Some applications of a criterion for Artinianness of a module , J. Pure Appl.Algebra (1995), 291-303.[12] L. Melkersson,
Modules cofinite with respect to an ideal , J. Algebra (2005), 649-668.[13] Sh. Payrovi, M. Lotfi Parsa,
Finiteness of local cohomology modules defined by a pair ofideals , Comm. Algebra (2013), 627-637.[14] Sh. Payrovi, M. Lotfi Parsa, Regular sequences and ZD-modules , Eprint arXiv: 1305.0164v1.[15] J. Rotman,
An Introduction to Homological Algebra , Academic Press, Orlando, 1979.[16] R. Takahashi, Y. Yoshino, T. Yoshizawa,
Local cohomology based on a nonclosed supportdefined by a pair of ideals , J. Pure Appl. Algebra (2009), 582-600.[17] A. Tehranian, A. Pour Eshmanan Talemi,
Cofiniteness of local cohomology based on a non-closed support defined by a pair of ideals , Bull. Iranian Math. Soc. (2010), 145-155.I. K. I NTERNATIONAL U NIVERSITY , P
OSTAL C ODE : 34149-1-6818 Q
AZVIN - IRAN
E-mail address : [email protected] E-mail address ::